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A239909 Arises from a construction of equiangular lines in complex space of dimension 2.

%I A239909 #29 Jan 29 2025 19:47:32
%S A239909 1,1,2,3,5,9,15,26,45,77,133,229,394,679,1169,2013,3467,5970,10281,
%T A239909 17705,30489,52505,90418,155707,268141,461761,795191,1369386,2358197,
%U A239909 4061013,6993405,12043229,20739450,35715071,61504345,105915637,182395603,314100514
%N A239909 Arises from a construction of equiangular lines in complex space of dimension 2.
%C A239909 a(n+2) is the number of binary words of length n in which every run of zeros has length congruent to 1 modulo 3. - _Ira M. Gessel_, Jan 22 2025
%H A239909 Vincenzo Librandi, <a href="/A239909/b239909.txt">Table of n, a(n) for n = 1..1000</a>
%H A239909 G. McConnell, <a href="http://arxiv.org/abs/1402.7330">Some non-standard ways to generate SIC-POVMs in dimensions 2 and 3</a>, arXiiv preprint arXiv:1402.7330, 2014, p. 4.
%H A239909 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,1,-1).
%F A239909 From _Vincenzo Librandi_ Apr 09 2014: (Start)
%F A239909 G.f.: x*(1-x^3)/(x^4-x^3-x^2-x+1).
%F A239909 a(n) = a(n-1) + a(n-2) + a(n-3) - a(n-4) for n>4.
%F A239909 a(n) = a(n-1) + 2*a(n-3) + A116732(n-5) for n>4. (End)
%t A239909 LinearRecurrence[{1, 1, 1, -1}, {1, 1, 2, 3}, 40] (* or *) CoefficientList[Series[(1 - x^3)/(x^4 - x^3 - x^2 - x + 1), {x, 0, 100}], x] (* _Vincenzo Librandi_, Apr 09 2014 *)
%o A239909 (Magma) I:=[1,1,2,3]; [n le 4 select I[n] else Self(n-1)+Self(n-2)+Self(n-3)-Self(n-4): n in [1..50]];
%Y A239909 Cf. A116732.
%K A239909 nonn,easy
%O A239909 1,3
%A A239909 _N. J. A. Sloane_, Apr 09 2014